Volumetric lat map

ABSTRACT

A method includes assigning, to first voxels in a model of tissue of a chamber of a heart, respective first values of a parameter at respective locations on the tissue, the first voxels representing the locations, respectively. Some of the locations are on an endocardial surface of the tissue, and others of the locations are on an epicardial surface of the tissue. The method further includes assigning respective second values to second voxels in the model, a subset of which represent a portion of the tissue between the endocardial surface and the epicardial surface, by interpolating the first values. Other embodiments are also described.

CROSS-REFERENCE TO RELATED APPLICATION

The present application claims the benefit of U.S. Provisional Appl. No.62/852,266, entitled “Volumetric LAT map,” filed May 23, 2019, whosedisclosure is incorporated herein by reference.

FIELD OF THE INVENTION

The present invention is related to anatomical and electrophysiologicalmodels, particularly of the heart.

BACKGROUND

A local activation time (LAT) at a particular location on the tissue ofa heart is the time at which the wavefront of electrical propagationpasses through the location. A local activation time is typicallymeasured from a particular reference time, such as a point in time inthe QRS complex of a body-surface electrocardiogram (ECG) recording.

US Patent Application Publication 2006/0084970 describes a method ofacquiring and mapping physiological data in a heart chamber. The methodincludes inserting a catheter having an electrode into the heartchamber. Physiological data in the heart chamber is acquired with theelectrode. The position of the electrode is determined, and the locationof the acquired physiological data is determined using the position ofthe electrode. The acquired physiological data is integrated with thelocation of the acquired physiological data. Information related to thethree-dimensional geometry of at least a portion of the heart chamber isreceived, and a continuous three-dimensional color-coded map of thephysiological data is created and superimposed on a geometricalrepresentation of the three-dimensional geometry information. The map isthen utilized to deliver ablation therapy.

US Patent Application Publication 2016/0100770 describes a system fordiagnosing arrhythmias and directing catheter therapies, which may allowfor measuring, classifying, analyzing, and mapping spatialelectrophysiological (EP) patterns within a body. The system may furtherguide arrhythmia therapy and update maps as treatment is delivered. Thesystem may use a medical device having a high density of sensors with aknown spatial configuration for collecting EP data and positioning data.Further, the system may also use an electronic control system (ECU) forcomputing and providing the user with a variety of metrics, derivativemetrics, high definition (HD) maps, HD composite maps, and generalvisual aids for association with a geometrical anatomical model shown ona display device.

SUMMARY OF THE INVENTION

There is provided, in accordance with some embodiments of the presentinvention, a system including a monitor and a processor. The processoris configured to assign, to first voxels in a model of tissue of achamber of a heart, respective first values of a parameter at respectivelocations on the tissue, the first voxels representing the locations,respectively. Some of the locations are on an endocardial surface of thetissue, and others of the locations are on an epicardial surface of thetissue. The processor is further configured to assign respective secondvalues to second voxels in the model, a subset of which represent aportion of the tissue between the endocardial surface and the epicardialsurface, by interpolating the first values. The processor is furtherconfigured to display the model on the monitor.

In some embodiments, the parameter includes a property of the tissue.

In some embodiments, the property includes a local activation time(LAT).

In some embodiments, the processor is further configured to:

identify, based on the first values and the second values, at least oneregion of decelerating electrical propagation, and

generate an output indicating the region.

In some embodiments, the parameter includes an amount of energydelivered to the tissue.

In some embodiments, the processor is configured to interpolate thefirst values by iteratively assigning, to each voxel of the secondvoxels, an average of immediate neighbors of the voxel.

In some embodiments, the processor is configured to assign the averageby assigning a weighted average in which the immediate neighbors areweighted by respective weights, which are derived from respective levelsof confidence associated with the first values.

In some embodiments, the processor is configured to interpolate thefirst values by, prior to iteratively assigning the average to eachvoxel of the second voxels, assigning a respective initial value to eachvoxel of the second voxels, using any type of nearest neighborinterpolation.

In some embodiments, the processor is configured to display the model soas to indicate those of the second values assigned to the subset.

There is further provided, in accordance with some embodiments of thepresent invention, a method including assigning, to first voxels in amodel of tissue of a chamber of a heart, respective first values of aparameter at respective locations on the tissue, the first voxelsrepresenting the locations, respectively. Some of the locations are onan endocardial surface of the tissue, and others of the locations are onan epicardial surface of the tissue. The method further includesassigning respective second values to second voxels in the model, asubset of which represent a portion of the tissue between theendocardial surface and the epicardial surface, by interpolating thefirst values.

There is further provided, in accordance with some embodiments of thepresent invention, a computer software product including a tangiblenon-transitory computer-readable medium in which program instructionsare stored. The instructions, when read by a processor, cause theprocessor to assign, to first voxels in a model of tissue of a chamberof a heart, respective first values of a parameter at respectivelocations on the tissue, the first voxels representing the locations,respectively. Some of the locations are on an endocardial surface of thetissue, and others of the locations are on an epicardial surface of thetissue. The instructions further cause the processor to assignrespective second values to second voxels in the model, a subset ofwhich represent a portion of the tissue between the endocardial surfaceand the epicardial surface, by interpolating the first values.

The present invention will be more fully understood from the followingdetailed description of embodiments thereof, taken together with thedrawings, in which:

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic illustration of a system for generating anaugmented model of cardiac tissue, in accordance with some embodimentsof the present invention;

FIG. 2 is a flow diagram for a technique for augmenting a model ofcardiac tissue, in accordance with some embodiments of the presentinvention; and

FIG. 3 illustrates aspects of a technique for augmenting a model ofcardiac tissue, in accordance with some embodiments of the presentinvention.

DETAILED DESCRIPTION OF EMBODIMENTS Overview

In some applications, an electrophysiological map of a portion of asubject's heart is constructed. The electrophysiological map includes acomputerized representation of the anatomy of the portion of the heart,along with superimposed electrophysiological data. An example of such amap is an LAT map, which indicates respective LAT values at variousanatomical locations using, for example, a sliding color scale.

To construct an LAT map, one or more electrodes at the distal end of acatheter first acquire a “point cloud” of LAT values for variouslocations on the tissue of the heart. This point cloud is then mapped toa voxelized anatomical model of the tissue, such that the acquired LATvalues are assigned, respectively, to a subset of the voxels in themodel. Subsequently, using a suitable interpolation technique, theremaining voxels are assigned interpolated LAT values.

Per conventional techniques, two entirely independent LAT maps areconstructed for the endocardial and epicardial surfaces of the relevantportion of the heart, respectively. The present inventors have realized,however, that the electrophysiological properties of the endocardialsurface of the heart are correlated with those of the epicardialsurface, due to the propagation of electrical current through theinternal, or “intramural,” cardiac tissue. Hence, by virtue ofconsidering each surface in isolation, the aforementioned conventionaltechniques may provide inaccurate interpolated LAT values.

Hence, embodiments of the present invention provide a volumetric, i.e.,three-dimensional, LAT map, which considers both the endocardial andepicardial surfaces, along with the intramural tissue. First, athree-dimensional anatomical model of the cardiac tissue—including theendocardial surface, the epicardial surface, and the intramuraltissue—is constructed. Next, respective LAT point clouds are acquiredfor the endocardial and epicardial surfaces, and the LAT point cloudsare mapped to the model. Subsequently, using a suitable interpolationtechnique, LAT values are estimated for the remaining surface andintramural voxels. For example, an iterative interpolation technique maybe used, whereby, during each iteration, each voxel is assigned theaverage of the values of its immediate neighbors.

Advantageously, the volumetric map that is constructed as describedherein is generally accurate for both the endocardial and epicardialsurfaces. Moreover, the volumetric map may allow the physician tovisualize the electrophysiological properties of the intramural tissue.Furthermore, the volumetric map may facilitate a more accurateidentification of regions at which the electrical propagationdecelerates.

In addition to local activation times, the techniques described hereinmay be used to construct volumetric maps for other parameters associatedwith the tissue. Such parameters include voltage, cycle length,temperature, and an amount of energy delivered to the tissue.

System Description

Reference is initially made to FIG. 1 , which is a schematicillustration of a system 20 for generating an augmented model of cardiactissue, in accordance with some embodiments of the present invention.

In FIG. 1 , a physician 30 is shown moving the distal end 28 of acatheter 26 along tissue of a chamber of the heart 24 of a subject 22.In particular, physician 30 moves distal end 28 along both theendocardial surface and epicardial surface of the tissue. In someembodiments, distal end 28 includes one or more treatment electrodes,which may be used to treat (e.g., ablate) one or more areas of tissue.

While the distal end of the catheter is moved along the tissue, aprocessor 32 belonging to system 20 tracks the distal end, i.e.,ascertains the multiple locations on the tissue at which distal end 28is disposed. (For convenience, each of these locations is referred tohereinbelow simply as the location of the catheter.) As noted above,some of these locations are on the endocardial surface of the tissue,while others are on the epicardial surface of the tissue.

In addition, while the distal end of the catheter is moved along thetissue, electrodes and/or other sensors (e.g., temperature or forcesensors) disposed at distal end 28 acquire data related to at least oneparameter. These data are received by processor 32 via an electricalinterface 34, such as a port or socket. Based on these data, processor32 ascertains respective values of the parameter at the multiplelocations.

Typically, the data acquired by distal end 28 include respective voltagesignals at the various locations of the tissue over which distal end 28is passed. Alternatively or additionally, the data may includerespective temperature values at the locations. Alternatively oradditionally, the data may include the force with which the catheterpresses against the tissue.

In some embodiments, based on the locations of the catheter that areascertained, processor 32 constructs an anatomical model of the tissue.This anatomical model is then augmented with the values of theaforementioned parameter, as further described below with reference toFIGS. 2-3 . In other embodiments, processor 32 augments a preexistinganatomical model with the parameter values.

To facilitate tracking the distal end of the catheter, the distal end ofthe catheter may comprise one or more electromagnetic sensors, which, inthe presence of a generated magnetic field, output signals indicatingthe respective locations of the sensors. These signals may be receivedby processor 32 via electrical interface 34. Based on the signals,processor 32 may ascertain the location of the catheter.

Alternatively, the distal end of the catheter may comprise a catheterelectrode, and a plurality of electrode patches may be coupled to thebody of subject 22. As voltages are applied between the catheterelectrode and the electrode patches, the respective magnitudes of thecurrents between the catheter electrode and the electrode patches may bemeasured. Based on these current magnitudes, the processor may ascertainthe location of the catheter.

As yet another alternative, both of the above-described trackingtechniques may be used in combination with one another, as described,for example, in U.S. Pat. No. 8,456,182, whose disclosure isincorporated herein by reference. Alternatively or additionally, anyother suitable tracking technique may be used, e.g., as described inU.S. Pat. No. 8,456,182.

Typically, system 20 further comprises a monitor 36. As the physicianoperates catheter 26, processor 32 may superimpose, on monitor 36, anicon representing the distal end of the catheter over an image of thesubject's heart, such that the physician may visually track thecatheter. Alternatively or additionally, the processor may display anaugmented model of the tissue, which may be constructed as described indetail hereinbelow with reference to FIGS. 2-3 , on monitor 36.

In general, processor 32 may be embodied as a single processor, or as acooperatively networked or clustered set of processors. In someembodiments, the functionality of processor 32, as described herein, isimplemented solely in hardware, e.g., using one or moreApplication-Specific Integrated Circuits (ASICs) or Field-ProgrammableGate Arrays (FPGAs). In other embodiments, the functionality ofprocessor 32 is implemented at least partly in software. For example, insome embodiments, processor 32 is a programmed digital computing devicecomprising a central processing unit (CPU) and/or a graphics processingunit (GPU), random access memory (RAM), non-volatile secondary storage,such as a hard drive or CD ROM drive, network interfaces, and/orperipheral devices. Program code, including software programs, and/ordata are loaded into the RAM for execution and processing by the CPUand/or GPU, and results are generated for display, output, transmittal,or storage, as is known in the art. The program code and/or data may bedownloaded to the computer in electronic form, over a network, forexample, or it may, alternatively or additionally, be provided and/orstored on non-transitory tangible media, such as magnetic, optical, orelectronic memory. Such program code and/or data, when provided to theprocessor, produce a machine or special-purpose computer, configured toperform the tasks described herein.

Augmenting the Model

Reference is now made to FIG. 2 , which is a schematic illustration of aflow diagram for a technique 48 for augmenting a model of cardiactissue, in accordance with some embodiments of the present invention.Reference is further made to FIG. 3 , which illustrates aspects oftechnique 48. (It is noted that the quantities shown in FIG. 3 arepurely hypothetical, for the sake of illustration.)

As described above with reference to FIG. 1 , at an ascertaining step50, processor 32 ascertains the respective values of a particularparameter at multiple locations on the endocardial and epicardialsurfaces of a chamber of heart 24. For example, the processor mayascertain respective values of a property of the tissue, such as thevoltage, LAT, cycle-length, or temperature of the tissue. (LAT andcycle-length values may be derived from the voltage signals acquiredfrom the tissue.) Alternatively or additionally, the processor mayascertain respective values of an amount of energy, such asradiofrequency (RF) energy, delivered to the tissue by the treatmentelectrodes. The amount of energy may be calculated based on factors suchas the amount of energy delivered to the treatment electrodes, thetemperature of the tissue, and the pressure with which the catheterpresses against the tissue.

Subsequently, the processor associates the ascertained values with athree-dimensional model 38 of the tissue. Model 38 includes a pluralityof voxels, each voxel representing a different respective portion of thetissue. In particular, those voxels that define a first surface 40 ofthe model—referred to herein as “epicardial voxels”—represent theepicardial surface of the tissue, those voxels that define a secondsurface 42—referred to herein as “endocardial voxels”—represent theendocardial surface, and those voxels lying between first surface 40 andsecond surface 42—referred to herein as “intramural voxels”—representthe intramural tissue. Those voxels representing the respective portionsof tissue at which the ascertained parameter values were exhibited arereferred to herein as first voxels 44.

More specifically, at a first assigning step 52, the values of theparameter that were ascertained at ascertaining step 50 are assigned tofirst voxels 44, respectively, as shown in section A of FIG. 3 . Inother words, each first voxel 44 is assigned the value that wasexhibited at the portion of tissue represented by the voxel. Next, asshown in sections B-D of FIG. 3 , the processor values (i.e., assignsrespective values to) the remaining voxels, referred to herein as secondvoxels 46, by interpolating the values assigned to first voxels 44. (Forclarity, FIG. 3 italicizes the values assigned to second voxels 46.)

Typically, to value the second voxels, the processor first initializesthe second voxels at an initializing step 54, i.e., the processorassigns a respective initial value to each second voxel 46. To performthis initialization, the processor may use any suitable type of nearestneighbor interpolation. For example, as shown in section B of FIG. 3 ,the processor may use a standard nearest neighbor interpolationtechnique, in that each second voxel may be assigned the value of thefirst voxel nearest to it. Alternatively, for example, a weightednearest neighbor interpolation technique may be used for thisinitialization.

Typically, following the initialization, the processor iterativelyassigns, to each second voxel, the average of the respective values ofthe immediate neighbors of the second voxel. (For embodiments in whichthe above-described initialization is not performed, the average isperformed only over those immediate neighbors to which values werealready assigned.) This iterative averaging may be referred to as“Laplace interpolation.”

In some embodiments, the number of iterations is predefined. In otherembodiments, the processor performs the iterative averaging until one ormore predefined stopping criteria are satisfied. For example, theiterative averaging may be performed until the maximum differencebetween any neighboring pair of voxels is less than a predefinedthreshold.

Thus, for example, as shown in FIG. 2 , the iterative averaging maycomprise a second assigning step 56 and a checking step 58. At secondassigning step 56, the processor assigns, to each second voxel, theaverage of its immediate neighbors. At checking step 58, the processorchecks whether the predefined number of iterations have been performed,or whether the stopping criteria have been satisfied. If yes, theiterative averaging ends. Otherwise, the processor returns to secondassigning step 56.

In some embodiments, one voxel is considered to be an immediate neighborof (or “adjacent to”) another voxel if the two voxels share at least onevertex. Thus, a voxel may have up to 26 immediate neighbors. (Due to thetwo-dimensional representation of the voxels, FIG. 3 shows a maximum ofeight, rather than 26, immediate neighbors.) In other embodiments, twovoxels are considered to be immediate neighbors of one another only ifthe two voxels share at least one face; thus, a voxel may have a maximumof only six immediate neighbors. Alternatively, other criteria may beused for determining the immediate neighbors of a voxel.

By way of illustration, sections C and D of FIG. 3 show two iterationsof the above-described averaging, assuming that a pair of voxels sharingat least one common vertex are considered to be immediate neighbors ofone another. (It is noted that FIG. 3 does not consider any voxels thatare not fully shown in the figure; thus, for example, a corner voxel isvalued by averaging only three immediate neighbors.)

In some embodiments, in valuing each second voxel, the immediateneighbors of the second voxel are equally weighted, as assumed in FIG. 3. In other embodiments, the averaged values are weighted by respectiveweights, which are derived from respective levels of confidenceassociated with the values assigned to first voxels 44. These levels ofconfidence are generally a function of the quality with which therelevant data are received from the distal end of the catheter.

For example, supposing that the levels of confidence were greater forthe epicardial surface (represented by first surface 40) than for theendocardial surface, the processor might give a greater weight to eachepicardial first voxel, along with each “child” second voxel initializedto a value of an epicardial first voxel. Thus, for example, assuming aweight of 1.2 for each epicardial first voxel and the children thereof,and a weight of only 1 for each endocardial first voxel and the childrenthereof, the particular second voxel 46 a shown in section C would beassigned a value of 103.3 (=(1.2*500+330)/(1.2*5+3)), rather than 103.8.

Alternatively or additionally to Laplace interpolation, otherinterpolation techniques may be used to value second voxels 46. Suchtechniques include, for example, kriging, inverse distance weighting,spline interpolation, natural neighbor interpolation, and—as alreadydescribed above—nearest neighbor interpolation. In general, theinterpolation techniques are selected responsively to the properties ofthe interpolated parameter. For example, for local activation times,which vary linearly across the tissue, a linear interpolation technique,such as Laplace interpolation, may be used. For delivered energy, on theother hand, a non-linear, thermodynamics-based interpolation techniquemay be used. For example, the processor may assume that the amount ofdelivered energy decays exponentially from the site at which thetreatment electrodes contact the tissue.

In some embodiments, second voxels 46 are valued using multiple parallelexecution threads running, for example, on a graphics processing unit(GPU). Thus, for example, during each iteration of a Laplaceinterpolation, all of the second voxels may be processed in parallel.

In some cases, voxels corresponding to scar tissue are not valued, anddo not contribute to the valuing of other voxels. Scar tissue may beidentified manually by a physician or automatically by processor 32,based on the voltage signals acquired from the tissue.

Typically, subsequently to valuing the second voxels, the processordisplays model 38 on monitor 36 (FIG. 1 ) at a displaying step 60, so asto indicate the values of the parameter. For example, the processor maycolor the voxels of the model in accordance with a color scalecorresponding to the range of values attained by the parameter. As notedabove in the Overview, in displaying the model the processor typicallyindicates the values assigned to the intramural voxels. Thus,advantageously, the physician may obtain a better understanding of theelectrical properties of the tissue, relative to if only the endocardialand epicardial surfaces were to be shown.

In some embodiments, based on LAT values assigned to model 38, theprocessor identifies any regions of decelerating electrical propagation.Advantageously, the three-dimensional nature of model 38 facilitatesidentifying these regions with greater accuracy.

For example, at each voxel having the coordinates (x0,y0,z0), theprocessor may compute a normalized velocity of electrical propagation asV_((x0,y0,z0))=((L_((x0+1,y0,z0))−L_((x0−1,y0,z0)))⁻¹,(L_((x0,y0+1,z0))−L_((x0,y0−1,z0)))⁻¹,(L_((x0,y0,z0+1))−L_((x0,y0,z0−1)))⁻¹), where L_((x,y,z)) indicates theLAT at the voxel having the coordinates (x,y,z), (x0±1,y0,z0) are theimmediate neighbors of the voxel along the x-axis, (x0,y0±1,z0) are theimmediate neighbors of the voxel along the y-axis, and (x0,y0,z0±1) arethe immediate neighbors of the voxel along the z-axis. The processor maythen compute the derivative of the velocity asdV=(V_((x0+1,y0,z0))−V_((x0−1,y0,z0))),V_((x0,y0+1,z0))−V_((x0,y0−1,z0)), V_((x0,y0,z0+1))−V_((x0,y0,z0−1))).Subsequently, the processor may compute the dot product V·dV. If thisdot product is negative, the voxel is assumed to represent part of aregion of decelerating electrical propagation.

In response to identifying at least one region of deceleratingelectrical propagation, the processor may generate an output indicatingthe region. For example, in displaying the model, the processor maycolor or otherwise annotate the voxels that represent the region.

It will be appreciated by persons skilled in the art that the presentinvention is not limited to what has been particularly shown anddescribed hereinabove. Rather, the scope of embodiments of the presentinvention includes both combinations and subcombinations of the variousfeatures described hereinabove, as well as variations and modificationsthereof that are not in the prior art, which would occur to personsskilled in the art upon reading the foregoing description. Documentsincorporated by reference in the present patent application are to beconsidered an integral part of the application except that to the extentany terms are defined in these incorporated documents in a manner thatconflicts with the definitions made explicitly or implicitly in thepresent specification, only the definitions in the present specificationshould be considered.

1-20. (canceled)
 21. A system, comprising: a monitor; and a processor,configured to: assign, to first locations in a model of tissue of achamber of a heart, respective first values of a parameter, some of thefirst locations being on an endocardial surface of the tissue, andothers of the first locations being on an epicardial surface of thetissue, assign respective second values to second locations in themodel, a subset of which represent a portion of the tissue between theendocardial surface and the epicardial surface, by interpolating valuesbased on values assigned to respective neighbors of the secondlocations, and display the model on the monitor.
 22. The systemaccording to claim 21, wherein the parameter includes a property of thetissue.
 23. The system according to claim 22, wherein the propertyincludes a local activation time (LAT).
 24. The system according toclaim 23, wherein the processor is further configured to: identify,based on the first values and the second values, at least one region ofdecelerating electrical propagation, and generate an output indicatingthe region.
 25. The system according to claim 21, wherein the parameterincludes an amount of energy delivered to the tissue.
 26. The systemaccording to claim 21, wherein the processor is configured tointerpolate values based on values assigned to respective neighbors ofthe second locations by iteratively assigning, to each location of thesecond locations, an average of values assigned to immediate neighborsof the location.
 27. The system according to claim 26, wherein theprocessor is configured to assign the average by assigning a weightedaverage in which the immediate neighbors are weighted by respectiveweights, which are derived from respective levels of confidenceassociated with the first values.
 28. The system according to claim 26,wherein the processor is configured to interpolate values based onvalues assigned to respective neighbors of the second locations by,prior to iteratively assigning the average to each location of thesecond locations, assigning a respective initial value to each locationof the second locations, using any type of nearest neighborinterpolation.
 29. The system according to claim 21, wherein theprocessor is configured to display the model so as to indicate those ofthe second values assigned to the subset.
 30. A method, comprising:assigning, to first locations in a model of tissue of a chamber of aheart, respective first values of a parameter at respective locations onthe tissue, some of the first locations being on an endocardial surfaceof the tissue, and others of the first locations being on an epicardialsurface of the tissue; and assigning respective second values to secondlocations in the model, a subset of which represent a portion of thetissue between the endocardial surface and the epicardial surface, byinterpolating values based on values assigned to respective neighbors ofthe second locations.
 31. The method according to claim 30, wherein theparameter includes a property of the tissue.
 32. The method according toclaim 31, wherein the property includes a local activation time (LAT).33. The method according to claim 32, further comprising: based on thefirst values and the second values, identifying at least one region ofdecelerating electrical propagation; and generating an output indicatingthe region.
 34. The method according to claim 30, wherein the parameterincludes an amount of energy delivered to the tissue.
 35. The methodaccording to claim 30, wherein interpolating values based on valuesassigned to respective neighbors of the second locations iterativelyassigning, to each voxel of the second voxels, an average of immediateneighbors of the voxel.
 36. The method according to claim 35, whereinassigning the average comprises assigning a weighted average in whichthe immediate neighbors are weighted by respective weights, which arederived from respective levels of confidence associated with the firstvalues.
 37. The method according to claim 35, wherein interpolatingvalues based on values assigned to respective neighbors of the secondlocations further comprises, prior to iteratively assigning the averageto each location of the second locations, assigning a respective initialvalue to each location of the second locations, using any type ofnearest neighbor interpolation.
 38. The method according to claim 30,further comprising displaying the model so as to indicate those of thesecond values assigned to the subset.
 39. A computer software productcomprising a tangible non-transitory computer-readable medium in whichprogram instructions are stored, which instructions, when read by aprocessor, cause the processor to: assign, to first locations in a modelof tissue of a chamber of a heart, respective first values of aparameter at respective locations on the tissue, some of the firstlocations being on an endocardial surface of the tissue, and others ofthe first locations being on an epicardial surface of the tissue, andassign respective second values to second locations in the model, asubset of which represent a portion of the tissue between theendocardial surface and the epicardial surface, by interpolating valuesbased on values assigned to respective neighbors of the secondlocations.
 40. The computer software product according to claim 39,wherein the parameter includes a local activation time (LAT) of thetissue.